Question

The condition for which the eigen values of the matrix A = $\left[\begin{array}{cc}2& 1\\ 1& k\end{array}\right]$ are positive, is

• A. k > 1/2
• B. k > -2
• C. k > 0
• D. k < - 1/2

Answer 1

The correct option is A k > 1/2

A = $\left[\begin{array}{cc}2& 1\\ 1& K\end{array}\right]$

$|A-\lambda I|$ = 0

$\Rightarrow$ $\left|\begin{array}{cc}2-\lambda & 1\\ 1& k-\lambda \end{array}\right|$ = 0

$\Rightarrow$ $(2-\lambda )(k-\lambda )-1=0$

$\Rightarrow$ ${\lambda }^2-\lambda (k+2)+(2k-1)=0$

For both values of '$\lambda$' to be positive, this implies to

${\lambda }_1+{\lambda }_2>0\Rightarrow 2+k>0\Rightarrow k>-2and{\lambda }_1.{\lambda }_2>0\Rightarrow 2k-1>0\Rightarrow k>\frac{1}{2}$

Now taking intersection of both inequalities, we get k> $\frac{1}{2}$